3.2.78 \(\int (a+b \tanh ^{-1}(\frac {c}{x^2}))^2 \, dx\) [178]

Optimal. Leaf size=1549 \[ \text {result too large to display} \]

[Out]

-1/2*b^2*ln(1+c/x^2)*ln(-x+(-c)^(1/2))*(-c)^(1/2)+1/2*b^2*ln(1+c/x^2)*ln(x+(-c)^(1/2))*(-c)^(1/2)-1/2*b^2*ln(1
/2*(-x+(-c)^(1/2))/(-c)^(1/2))*ln(x+(-c)^(1/2))*(-c)^(1/2)+1/2*b^2*ln(-x+(-c)^(1/2))*ln(1/2*(x+(-c)^(1/2))/(-c
)^(1/2))*(-c)^(1/2)+2*a*b*arctan(x/c^(1/2))*c^(1/2)-2*a*b*arctanh(x/c^(1/2))*c^(1/2)-1/2*b^2*ln(1-c/x^2)*ln(-x
+c^(1/2))*c^(1/2)-2*b^2*arctan(x/c^(1/2))*ln(2*c^(1/2)/(-I*x+c^(1/2)))*c^(1/2)-2*b^2*arctanh(x/c^(1/2))*ln(2*c
^(1/2)/(x+c^(1/2)))*c^(1/2)+1/2*b^2*ln(1-c/x^2)*ln(x+c^(1/2))*c^(1/2)-1/2*b^2*ln(1/2*(-x+c^(1/2))/c^(1/2))*ln(
x+c^(1/2))*c^(1/2)+1/2*b^2*ln(-x+c^(1/2))*ln(1/2*(x+c^(1/2))/c^(1/2))*c^(1/2)-I*b^2*polylog(2,-I*x/c^(1/2))*c^
(1/2)-1/2*I*b^2*polylog(2,1-(1+I)*(-x+c^(1/2))/(-I*x+c^(1/2)))*c^(1/2)-1/2*I*b^2*polylog(2,1+(-1+I)*(x+c^(1/2)
)/(-I*x+c^(1/2)))*c^(1/2)-a*b*x*ln(1-c/x^2)-b^2*ln(x/(-c)^(1/2))*ln(-x+(-c)^(1/2))*(-c)^(1/2)+a*b*x*ln(1+c/x^2
)+b^2*ln(-x/(-c)^(1/2))*ln(x+(-c)^(1/2))*(-c)^(1/2)-b^2*arctan(x/c^(1/2))*ln(1-c/x^2)*c^(1/2)-b^2*arctanh(x/c^
(1/2))*ln(1+c/x^2)*c^(1/2)-b^2*ln(x/c^(1/2))*ln(-x+c^(1/2))*c^(1/2)+b^2*arctan(x/c^(1/2))*ln((1+I)*(-x+c^(1/2)
)/(-I*x+c^(1/2)))*c^(1/2)+b^2*arctanh(x/c^(1/2))*ln(2*(-x+(-c)^(1/2))*c^(1/2)/((-c)^(1/2)-c^(1/2))/(x+c^(1/2))
)*c^(1/2)+b^2*ln(-x/c^(1/2))*ln(x+c^(1/2))*c^(1/2)+b^2*arctan(x/c^(1/2))*ln((1-I)*(x+c^(1/2))/(-I*x+c^(1/2)))*
c^(1/2)+b^2*arctanh(x/c^(1/2))*ln(2*(x+(-c)^(1/2))*c^(1/2)/(x+c^(1/2))/((-c)^(1/2)+c^(1/2)))*c^(1/2)+I*b^2*pol
ylog(2,I*x/c^(1/2))*c^(1/2)+I*b^2*polylog(2,1-2*c^(1/2)/(-I*x+c^(1/2)))*c^(1/2)-1/2*b^2*x*ln(1-c/x^2)*ln(1+c/x
^2)+1/4*b^2*ln(-x+c^(1/2))^2*c^(1/2)-1/4*b^2*ln(x+c^(1/2))^2*c^(1/2)+1/2*b^2*polylog(2,1/2-1/2*x/c^(1/2))*c^(1
/2)-1/2*b^2*polylog(2,1/2*(x+c^(1/2))/c^(1/2))*c^(1/2)-1/2*b^2*polylog(2,1-2*(-x+(-c)^(1/2))*c^(1/2)/((-c)^(1/
2)-c^(1/2))/(x+c^(1/2)))*c^(1/2)-1/2*b^2*polylog(2,1-2*(x+(-c)^(1/2))*c^(1/2)/(x+c^(1/2))/((-c)^(1/2)+c^(1/2))
)*c^(1/2)-b^2*polylog(2,1-x/(-c)^(1/2))*(-c)^(1/2)+b^2*polylog(2,1+x/(-c)^(1/2))*(-c)^(1/2)-b^2*polylog(2,1-x/
c^(1/2))*c^(1/2)+b^2*polylog(2,1+x/c^(1/2))*c^(1/2)+b^2*polylog(2,-x/c^(1/2))*c^(1/2)-b^2*polylog(2,x/c^(1/2))
*c^(1/2)+b^2*polylog(2,1-2*c^(1/2)/(x+c^(1/2)))*c^(1/2)+1/4*b^2*x*ln(1-c/x^2)^2+1/4*b^2*x*ln(1+c/x^2)^2+1/4*b^
2*ln(-x+(-c)^(1/2))^2*(-c)^(1/2)-1/4*b^2*ln(x+(-c)^(1/2))^2*(-c)^(1/2)+1/2*b^2*polylog(2,1/2-1/2*x/(-c)^(1/2))
*(-c)^(1/2)-1/2*b^2*polylog(2,1/2*(c-x*(-c)^(1/2))/c)*(-c)^(1/2)+a^2*x

________________________________________________________________________________________

Rubi [A]
time = 1.61, antiderivative size = 1549, normalized size of antiderivative = 1.00, number of steps used = 100, number of rules used = 30, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.500, Rules used = {6025, 6024, 2498, 269, 213, 2500, 2526, 2512, 266, 2463, 2441, 2352, 2437, 2338, 2440, 2438, 209, 2636, 12, 2520, 6820, 6139, 6031, 6057, 2449, 2497, 210, 5048, 4940, 4966} \begin {gather*} x a^2+2 b \sqrt {c} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) a-2 b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) a-b x \log \left (1-\frac {c}{x^2}\right ) a+b x \log \left (\frac {c}{x^2}+1\right ) a+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (\frac {c}{x^2}+1\right )+\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}-x\right )+\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}-x\right )-\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (x+\sqrt {-c}\right )-\frac {1}{4} b^2 \sqrt {c} \log ^2\left (x+\sqrt {c}\right )-b^2 \sqrt {c} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (\frac {c}{x^2}+1\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {c}{x^2}+1\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )-2 b^2 \sqrt {c} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {c}{x^2}+1\right ) \log \left (x+\sqrt {-c}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right ) \log \left (x+\sqrt {-c}\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (x+\sqrt {-c}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x+\sqrt {-c}}{2 \sqrt {-c}}\right )-2 b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (x+\sqrt {-c}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (x+\sqrt {c}\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right ) \log \left (x+\sqrt {c}\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (x+\sqrt {c}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x+\sqrt {c}}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right )+i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-i b^2 \sqrt {c} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+i b^2 \sqrt {c} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {x+\sqrt {c}}{2 \sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {1}{2} \left (1-\frac {x}{\sqrt {-c}}\right )\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (\frac {x}{\sqrt {-c}}+1\right )-\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {c-\sqrt {-c} x}{2 c}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {1}{2}-\frac {x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}+1\right )+b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (x+\sqrt {-c}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])^2,x]

[Out]

a^2*x + 2*a*b*Sqrt[c]*ArcTan[x/Sqrt[c]] - 2*a*b*Sqrt[c]*ArcTanh[x/Sqrt[c]] - a*b*x*Log[1 - c/x^2] - b^2*Sqrt[c
]*ArcTan[x/Sqrt[c]]*Log[1 - c/x^2] + (b^2*x*Log[1 - c/x^2]^2)/4 + a*b*x*Log[1 + c/x^2] - b^2*Sqrt[c]*ArcTanh[x
/Sqrt[c]]*Log[1 + c/x^2] - (b^2*x*Log[1 - c/x^2]*Log[1 + c/x^2])/2 + (b^2*x*Log[1 + c/x^2]^2)/4 - (b^2*Sqrt[-c
]*Log[1 + c/x^2]*Log[Sqrt[-c] - x])/2 + (b^2*Sqrt[-c]*Log[Sqrt[-c] - x]^2)/4 - (b^2*Sqrt[c]*Log[1 - c/x^2]*Log
[Sqrt[c] - x])/2 + (b^2*Sqrt[c]*Log[Sqrt[c] - x]^2)/4 - 2*b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[
c] - I*x)] + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] - b^2*Sqrt[-c]*Log[Sqr
t[-c] - x]*Log[x/Sqrt[-c]] - b^2*Sqrt[c]*Log[Sqrt[c] - x]*Log[x/Sqrt[c]] + (b^2*Sqrt[-c]*Log[1 + c/x^2]*Log[Sq
rt[-c] + x])/2 - (b^2*Sqrt[-c]*Log[(Sqrt[-c] - x)/(2*Sqrt[-c])]*Log[Sqrt[-c] + x])/2 + b^2*Sqrt[-c]*Log[-(x/Sq
rt[-c])]*Log[Sqrt[-c] + x] - (b^2*Sqrt[-c]*Log[Sqrt[-c] + x]^2)/4 + (b^2*Sqrt[-c]*Log[Sqrt[-c] - x]*Log[(Sqrt[
-c] + x)/(2*Sqrt[-c])])/2 - 2*b^2*Sqrt[c]*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)] + b^2*Sqrt[c]*ArcT
anh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))] + b^2*Sqrt[c]*ArcTanh[x/Sq
rt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c] + x))] + (b^2*Sqrt[c]*Log[1 - c/x^2]*Log[
Sqrt[c] + x])/2 - (b^2*Sqrt[c]*Log[(Sqrt[c] - x)/(2*Sqrt[c])]*Log[Sqrt[c] + x])/2 + b^2*Sqrt[c]*Log[-(x/Sqrt[c
])]*Log[Sqrt[c] + x] - (b^2*Sqrt[c]*Log[Sqrt[c] + x]^2)/4 + (b^2*Sqrt[c]*Log[Sqrt[c] - x]*Log[(Sqrt[c] + x)/(2
*Sqrt[c])])/2 + b^2*Sqrt[c]*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)] + I*b^2*Sqrt[c]*Pol
yLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)] - (I/2)*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] -
I*x)] + b^2*Sqrt[c]*PolyLog[2, -(x/Sqrt[c])] - I*b^2*Sqrt[c]*PolyLog[2, ((-I)*x)/Sqrt[c]] + I*b^2*Sqrt[c]*Pol
yLog[2, (I*x)/Sqrt[c]] - b^2*Sqrt[c]*PolyLog[2, x/Sqrt[c]] - (b^2*Sqrt[c]*PolyLog[2, (Sqrt[c] + x)/(2*Sqrt[c])
])/2 + (b^2*Sqrt[-c]*PolyLog[2, (1 - x/Sqrt[-c])/2])/2 - b^2*Sqrt[-c]*PolyLog[2, 1 - x/Sqrt[-c]] + b^2*Sqrt[-c
]*PolyLog[2, 1 + x/Sqrt[-c]] - (b^2*Sqrt[-c]*PolyLog[2, (c - Sqrt[-c]*x)/(2*c)])/2 - b^2*Sqrt[c]*PolyLog[2, 1
- x/Sqrt[c]] + (b^2*Sqrt[c]*PolyLog[2, 1/2 - x/(2*Sqrt[c])])/2 + b^2*Sqrt[c]*PolyLog[2, 1 + x/Sqrt[c]] + b^2*S
qrt[c]*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)] - (b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sq
rt[-c] - Sqrt[c])*(Sqrt[c] + x))])/2 - (b^2*Sqrt[c]*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqr
t[c])*(Sqrt[c] + x))])/2 - (I/2)*b^2*Sqrt[c]*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2500

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x^
n)^p])^q, x] - Dist[b*e*n*p*q, Int[x^n*((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a,
b, c, d, e, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])

Rule 2512

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[f +
g*x]*((a + b*Log[c*(d + e*x^n)^p])/g), x] - Dist[b*e*n*(p/g), Int[x^(n - 1)*(Log[f + g*x]/(d + e*x^n)), x], x]
/; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && RationalQ[n]

Rule 2520

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[u*(x^(n - 1)/(d + e*x^n)
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2636

Int[Log[v_]*Log[w_], x_Symbol] :> Simp[x*Log[v]*Log[w], x] + (-Int[SimplifyIntegrand[x*Log[w]*(D[v, x]/v), x],
x] - Int[SimplifyIntegrand[x*Log[v]*(D[w, x]/w), x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFree
Q[w, x]

Rule 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 4966

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x]))*(Log[2/(1
- I*c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((
d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], x] + Simp[(a + b*ArcTan[c*x])*(Log[2*c*((d + e*x)/((c*
d + I*e)*(1 - I*c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 5048

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 6024

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*(Log[1 + 1/(x^n*c)]
/2) - b*(Log[1 - 1/(x^n*c)]/2))^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rule 6025

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[(a + b*ArcCoth[1/(x^n*c)])^p, x] /; FreeQ[
{a, b, c}, x] && IGtQ[p, 1] && ILtQ[n, 0]

Rule 6031

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b/2)*PolyLog[2, (-c)*x]
, x] + Simp[(b/2)*PolyLog[2, c*x], x]) /; FreeQ[{a, b, c}, x]

Rule 6057

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x]))*(Log[2/
(1 + c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((d
+ e*x)/((c*d + e)*(1 + c*x)))]/(1 - c^2*x^2), x], x] + Simp[(a + b*ArcTanh[c*x])*(Log[2*c*((d + e*x)/((c*d + e
)*(1 + c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rule 6139

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {align*} \int \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx &=\int \left (a^2-a b \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 \log ^2\left (1-\frac {c}{x^2}\right )+a b \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 \log ^2\left (1+\frac {c}{x^2}\right )\right ) \, dx\\ &=a^2 x-(a b) \int \log \left (1-\frac {c}{x^2}\right ) \, dx+(a b) \int \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{4} b^2 \int \log ^2\left (1-\frac {c}{x^2}\right ) \, dx+\frac {1}{4} b^2 \int \log ^2\left (1+\frac {c}{x^2}\right ) \, dx-\frac {1}{2} b^2 \int \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right ) \, dx\\ &=a^2 x-a b x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{2} b^2 \int \frac {2 c \log \left (1-\frac {c}{x^2}\right )}{-c-x^2} \, dx+\frac {1}{2} b^2 \int \frac {2 c \log \left (1+\frac {c}{x^2}\right )}{-c+x^2} \, dx+(2 a b c) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^2} \, dx+(2 a b c) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^2} \, dx-\left (b^2 c\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{\left (1-\frac {c}{x^2}\right ) x^2} \, dx+\left (b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{\left (1+\frac {c}{x^2}\right ) x^2} \, dx\\ &=a^2 x-a b x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )+(2 a b c) \int \frac {1}{-c+x^2} \, dx+(2 a b c) \int \frac {1}{c+x^2} \, dx+\left (b^2 c\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{-c-x^2} \, dx-\left (b^2 c\right ) \int \left (-\frac {\log \left (1-\frac {c}{x^2}\right )}{2 \sqrt {c} \left (\sqrt {c}-x\right )}-\frac {\log \left (1-\frac {c}{x^2}\right )}{2 \sqrt {c} \left (\sqrt {c}+x\right )}\right ) \, dx+\left (b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{-c+x^2} \, dx+\left (b^2 c\right ) \int \left (\frac {\sqrt {-c} \log \left (1+\frac {c}{x^2}\right )}{2 c \left (\sqrt {-c}-x\right )}+\frac {\sqrt {-c} \log \left (1+\frac {c}{x^2}\right )}{2 c \left (\sqrt {-c}+x\right )}\right ) \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{\sqrt {-c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{\sqrt {-c}+x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{\sqrt {c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{\sqrt {c}+x} \, dx+\left (2 b^2 c^2\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1-\frac {c}{x^2}\right ) x^3} \, dx-\left (2 b^2 c^2\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1+\frac {c}{x^2}\right ) x^3} \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )+\left (b^2 (-c)^{3/2}\right ) \int \frac {\log \left (\sqrt {-c}-x\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx-\left (b^2 (-c)^{3/2}\right ) \int \frac {\log \left (\sqrt {-c}+x\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx+\left (b^2 c^{3/2}\right ) \int \frac {\log \left (\sqrt {c}-x\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx-\left (b^2 c^{3/2}\right ) \int \frac {\log \left (\sqrt {c}+x\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx+\left (2 b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx-\left (2 b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )+\left (b^2 (-c)^{3/2}\right ) \int \left (\frac {\log \left (\sqrt {-c}-x\right )}{c x}-\frac {x \log \left (\sqrt {-c}-x\right )}{c \left (c+x^2\right )}\right ) \, dx-\left (b^2 (-c)^{3/2}\right ) \int \left (\frac {\log \left (\sqrt {-c}+x\right )}{c x}-\frac {x \log \left (\sqrt {-c}+x\right )}{c \left (c+x^2\right )}\right ) \, dx+\left (b^2 c^{3/2}\right ) \int \left (-\frac {\log \left (\sqrt {c}-x\right )}{c x}-\frac {x \log \left (\sqrt {c}-x\right )}{c \left (c-x^2\right )}\right ) \, dx-\left (b^2 c^{3/2}\right ) \int \left (-\frac {\log \left (\sqrt {c}+x\right )}{c x}-\frac {x \log \left (\sqrt {c}+x\right )}{c \left (c-x^2\right )}\right ) \, dx+\left (2 b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (-c+x^2\right )} \, dx-\left (2 b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (c+x^2\right )} \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )-\left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}-x\right )}{x} \, dx+\left (b^2 \sqrt {-c}\right ) \int \frac {x \log \left (\sqrt {-c}-x\right )}{c+x^2} \, dx+\left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}+x\right )}{x} \, dx-\left (b^2 \sqrt {-c}\right ) \int \frac {x \log \left (\sqrt {-c}+x\right )}{c+x^2} \, dx-\left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}-x\right )}{x} \, dx-\left (b^2 \sqrt {c}\right ) \int \frac {x \log \left (\sqrt {c}-x\right )}{c-x^2} \, dx+\left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}+x\right )}{x} \, dx+\left (b^2 \sqrt {c}\right ) \int \frac {x \log \left (\sqrt {c}+x\right )}{c-x^2} \, dx+\left (2 b^2 c^{3/2}\right ) \int \left (-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c x}-\frac {x \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c \left (c-x^2\right )}\right ) \, dx-\left (2 b^2 c^{3/2}\right ) \int \left (\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c x}-\frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c \left (c+x^2\right )}\right ) \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+\left (b^2 \sqrt {-c}\right ) \int \left (-\frac {\log \left (\sqrt {-c}-x\right )}{2 \left (\sqrt {-c}-x\right )}+\frac {\log \left (\sqrt {-c}-x\right )}{2 \left (\sqrt {-c}+x\right )}\right ) \, dx-\left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (-\frac {x}{\sqrt {-c}}\right )}{\sqrt {-c}+x} \, dx-\left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\frac {x}{\sqrt {-c}}\right )}{\sqrt {-c}-x} \, dx-\left (b^2 \sqrt {-c}\right ) \int \left (-\frac {\log \left (\sqrt {-c}+x\right )}{2 \left (\sqrt {-c}-x\right )}+\frac {\log \left (\sqrt {-c}+x\right )}{2 \left (\sqrt {-c}+x\right )}\right ) \, dx-\left (b^2 \sqrt {c}\right ) \int \left (\frac {\log \left (\sqrt {c}-x\right )}{2 \left (\sqrt {c}-x\right )}-\frac {\log \left (\sqrt {c}-x\right )}{2 \left (\sqrt {c}+x\right )}\right ) \, dx-\left (b^2 \sqrt {c}\right ) \int \frac {\log \left (-\frac {x}{\sqrt {c}}\right )}{\sqrt {c}+x} \, dx-\left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}-x} \, dx+\left (b^2 \sqrt {c}\right ) \int \left (\frac {\log \left (\sqrt {c}+x\right )}{2 \left (\sqrt {c}-x\right )}-\frac {\log \left (\sqrt {c}+x\right )}{2 \left (\sqrt {c}+x\right )}\right ) \, dx-\left (2 b^2 \sqrt {c}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx-\left (2 b^2 \sqrt {c}\right ) \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c-x^2} \, dx-\left (2 b^2 \sqrt {c}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx+\left (2 b^2 \sqrt {c}\right ) \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c+x^2} \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (1+\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1+\frac {x}{\sqrt {c}}\right )-\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}-x\right )}{\sqrt {-c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}-x\right )}{\sqrt {-c}+x} \, dx+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}+x\right )}{\sqrt {-c}-x} \, dx-\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\sqrt {-c}+x\right )}{\sqrt {-c}+x} \, dx-\left (i b^2 \sqrt {c}\right ) \int \frac {\log \left (1-\frac {i x}{\sqrt {c}}\right )}{x} \, dx+\left (i b^2 \sqrt {c}\right ) \int \frac {\log \left (1+\frac {i x}{\sqrt {c}}\right )}{x} \, dx-\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}-x\right )}{\sqrt {c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}-x\right )}{\sqrt {c}+x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}+x\right )}{\sqrt {c}-x} \, dx-\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\sqrt {c}+x\right )}{\sqrt {c}+x} \, dx-\left (2 b^2 \sqrt {c}\right ) \int \left (\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {c}-x\right )}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {c}+x\right )}\right ) \, dx+\left (2 b^2 \sqrt {c}\right ) \int \left (-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {-c}-x\right )}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {-c}+x\right )}\right ) \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {\sqrt {-c}+x}{2 \sqrt {-c}}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-i b^2 \sqrt {c} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+i b^2 \sqrt {c} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (1+\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1+\frac {x}{\sqrt {c}}\right )+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (-\frac {-\sqrt {-c}-x}{2 \sqrt {-c}}\right )}{\sqrt {-c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \int \frac {\log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right )}{\sqrt {-c}+x} \, dx+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {-c}-x\right )-\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {-c}+x\right )+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (-\frac {-\sqrt {c}-x}{2 \sqrt {c}}\right )}{\sqrt {c}-x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \int \frac {\log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right )}{\sqrt {c}+x} \, dx+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {c}-x\right )-\frac {1}{2} \left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\sqrt {c}+x\right )-\left (b^2 \sqrt {c}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}-x} \, dx+\left (b^2 \sqrt {c}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}+x} \, dx-\left (b^2 \sqrt {c}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}-x} \, dx+\left (b^2 \sqrt {c}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}+x} \, dx\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )+\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}-x\right )-2 b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {\sqrt {-c}+x}{2 \sqrt {-c}}\right )-2 b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-i b^2 \sqrt {c} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+i b^2 \sqrt {c} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (1+\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1+\frac {x}{\sqrt {c}}\right )+2 \left (b^2 \int \frac {\log \left (\frac {2}{1-\frac {i x}{\sqrt {c}}}\right )}{1+\frac {x^2}{c}} \, dx\right )-b^2 \int \frac {\log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c} \left (1-\frac {i x}{\sqrt {c}}\right )}\right )}{1+\frac {x^2}{c}} \, dx-b^2 \int \frac {\log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c} \left (1-\frac {i x}{\sqrt {c}}\right )}\right )}{1+\frac {x^2}{c}} \, dx+2 \left (b^2 \int \frac {\log \left (\frac {2}{1+\frac {x}{\sqrt {c}}}\right )}{1-\frac {x^2}{c}} \, dx\right )-b^2 \int \frac {\log \left (\frac {2 \left (\sqrt {-c}-x\right )}{\left (-1+\frac {\sqrt {-c}}{\sqrt {c}}\right ) \sqrt {c} \left (1+\frac {x}{\sqrt {c}}\right )}\right )}{1-\frac {x^2}{c}} \, dx-b^2 \int \frac {\log \left (\frac {2 \left (\sqrt {-c}+x\right )}{\left (1+\frac {\sqrt {-c}}{\sqrt {c}}\right ) \sqrt {c} \left (1+\frac {x}{\sqrt {c}}\right )}\right )}{1-\frac {x^2}{c}} \, dx-\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {-c}}\right )}{x} \, dx,x,\sqrt {-c}-x\right )+\frac {1}{2} \left (b^2 \sqrt {-c}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {-c}}\right )}{x} \, dx,x,\sqrt {-c}+x\right )-\frac {1}{2} \left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {c}}\right )}{x} \, dx,x,\sqrt {c}-x\right )+\frac {1}{2} \left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 \sqrt {c}}\right )}{x} \, dx,x,\sqrt {c}+x\right )\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )+\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}-x\right )-2 b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {\sqrt {-c}+x}{2 \sqrt {-c}}\right )-2 b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-i b^2 \sqrt {c} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+i b^2 \sqrt {c} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {1}{2} \left (1-\frac {x}{\sqrt {-c}}\right )\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (1+\frac {x}{\sqrt {-c}}\right )-\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {c-\sqrt {-c} x}{2 c}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {1}{2}-\frac {x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1+\frac {x}{\sqrt {c}}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+2 \left (\left (i b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i x}{\sqrt {c}}}\right )\right )+2 \left (\left (b^2 \sqrt {c}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {x}{\sqrt {c}}}\right )\right )\\ &=a^2 x+2 a b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-2 a b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-a b x \log \left (1-\frac {c}{x^2}\right )-b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1-\frac {c}{x^2}\right )+a b x \log \left (1+\frac {c}{x^2}\right )-b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 x \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}-x\right )+\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}-x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}-x\right )+\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}-x\right )-2 b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {x}{\sqrt {-c}}\right )-b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (1+\frac {c}{x^2}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{2} b^2 \sqrt {-c} \log \left (\frac {\sqrt {-c}-x}{2 \sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )+b^2 \sqrt {-c} \log \left (-\frac {x}{\sqrt {-c}}\right ) \log \left (\sqrt {-c}+x\right )-\frac {1}{4} b^2 \sqrt {-c} \log ^2\left (\sqrt {-c}+x\right )+\frac {1}{2} b^2 \sqrt {-c} \log \left (\sqrt {-c}-x\right ) \log \left (\frac {\sqrt {-c}+x}{2 \sqrt {-c}}\right )-2 b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+b^2 \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (1-\frac {c}{x^2}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{2} b^2 \sqrt {c} \log \left (\frac {\sqrt {c}-x}{2 \sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )+b^2 \sqrt {c} \log \left (-\frac {x}{\sqrt {c}}\right ) \log \left (\sqrt {c}+x\right )-\frac {1}{4} b^2 \sqrt {c} \log ^2\left (\sqrt {c}+x\right )+\frac {1}{2} b^2 \sqrt {c} \log \left (\sqrt {c}-x\right ) \log \left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+b^2 \sqrt {c} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-i b^2 \sqrt {c} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+i b^2 \sqrt {c} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-b^2 \sqrt {c} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {\sqrt {c}+x}{2 \sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {1}{2} \left (1-\frac {x}{\sqrt {-c}}\right )\right )-b^2 \sqrt {-c} \text {Li}_2\left (1-\frac {x}{\sqrt {-c}}\right )+b^2 \sqrt {-c} \text {Li}_2\left (1+\frac {x}{\sqrt {-c}}\right )-\frac {1}{2} b^2 \sqrt {-c} \text {Li}_2\left (\frac {c-\sqrt {-c} x}{2 c}\right )-b^2 \sqrt {c} \text {Li}_2\left (1-\frac {x}{\sqrt {c}}\right )+\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (\frac {1}{2}-\frac {x}{2 \sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1+\frac {x}{\sqrt {c}}\right )+b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{2} b^2 \sqrt {c} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{2} i b^2 \sqrt {c} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )\\ \end {align*}

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Mathematica [A]
time = 2.76, size = 565, normalized size = 0.36 \begin {gather*} a^2 x-2 a b \sqrt {\frac {c}{x^2}} x \left (\text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )+\tanh ^{-1}\left (\sqrt {\frac {c}{x^2}}\right )\right )+2 a b x \tanh ^{-1}\left (\frac {c}{x^2}\right )-\frac {1}{2} b^2 \sqrt {\frac {c}{x^2}} x \left (-2 i \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )^2+4 \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right ) \tanh ^{-1}\left (\frac {c}{x^2}\right )-\frac {2 \tanh ^{-1}\left (\frac {c}{x^2}\right )^2}{\sqrt {\frac {c}{x^2}}}+2 \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right ) \log \left (1+e^{4 i \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )}\right )-2 \tanh ^{-1}\left (\frac {c}{x^2}\right ) \log \left (1-\sqrt {\frac {c}{x^2}}\right )+\log (2) \log \left (1-\sqrt {\frac {c}{x^2}}\right )-\frac {1}{2} \log ^2\left (1-\sqrt {\frac {c}{x^2}}\right )+\log \left (1-\sqrt {\frac {c}{x^2}}\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+\sqrt {\frac {c}{x^2}}\right )\right )+2 \tanh ^{-1}\left (\frac {c}{x^2}\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log (2) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log \left (\frac {1}{2} \left ((1+i)-(1-i) \sqrt {\frac {c}{x^2}}\right )\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {\frac {c}{x^2}}\right )\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )+\frac {1}{2} \log ^2\left (1+\sqrt {\frac {c}{x^2}}\right )+\log \left (1-\sqrt {\frac {c}{x^2}}\right ) \log \left (\frac {1}{2} \left ((1+i)+(1-i) \sqrt {\frac {c}{x^2}}\right )\right )-\frac {1}{2} i \text {PolyLog}\left (2,-e^{4 i \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )}\right )-\text {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {\frac {c}{x^2}}\right )\right )+\text {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {\frac {c}{x^2}}\right )\right )+\text {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {\frac {c}{x^2}}\right )\right )+\text {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {\frac {c}{x^2}}\right )\right )-\text {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {\frac {c}{x^2}}\right )\right )-\text {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {\frac {c}{x^2}}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c/x^2])^2,x]

[Out]

a^2*x - 2*a*b*Sqrt[c/x^2]*x*(ArcTan[Sqrt[c/x^2]] + ArcTanh[Sqrt[c/x^2]]) + 2*a*b*x*ArcTanh[c/x^2] - (b^2*Sqrt[
c/x^2]*x*((-2*I)*ArcTan[Sqrt[c/x^2]]^2 + 4*ArcTan[Sqrt[c/x^2]]*ArcTanh[c/x^2] - (2*ArcTanh[c/x^2]^2)/Sqrt[c/x^
2] + 2*ArcTan[Sqrt[c/x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c/x^2]])] - 2*ArcTanh[c/x^2]*Log[1 - Sqrt[c/x^2]] + Lo
g[2]*Log[1 - Sqrt[c/x^2]] - Log[1 - Sqrt[c/x^2]]^2/2 + Log[1 - Sqrt[c/x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c/x^2])
] + 2*ArcTanh[c/x^2]*Log[1 + Sqrt[c/x^2]] - Log[2]*Log[1 + Sqrt[c/x^2]] - Log[((1 + I) - (1 - I)*Sqrt[c/x^2])/
2]*Log[1 + Sqrt[c/x^2]] - Log[(-1/2 - I/2)*(I + Sqrt[c/x^2])]*Log[1 + Sqrt[c/x^2]] + Log[1 + Sqrt[c/x^2]]^2/2
+ Log[1 - Sqrt[c/x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c/x^2])/2] - (I/2)*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[c/x^2]]
)] - PolyLog[2, (1 - Sqrt[c/x^2])/2] + PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[c/x^2])] + PolyLog[2, (-1/2 + I/2)*(
-1 + Sqrt[c/x^2])] + PolyLog[2, (1 + Sqrt[c/x^2])/2] - PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c/x^2])] - PolyLog[2,
(1/2 + I/2)*(1 + Sqrt[c/x^2])]))/2

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c/x^2))^2,x)

[Out]

int((a+b*arctanh(c/x^2))^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x^2))^2,x, algorithm="maxima")

[Out]

(c*(2*arctan(x/sqrt(c))/sqrt(c) + log((x - sqrt(c))/(x + sqrt(c)))/sqrt(c)) + 2*x*arctanh(c/x^2))*a*b + 1/4*(x
*log(x^2 - c)^2 - integrate(-((x^2 - c)*log(x^2 + c)^2 - 2*(2*x^2 + (x^2 - c)*log(x^2 + c))*log(x^2 - c))/(x^2
- c), x))*b^2 + a^2*x

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x^2))^2,x, algorithm="fricas")

[Out]

integral(b^2*arctanh(c/x^2)^2 + 2*a*b*arctanh(c/x^2) + a^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c/x**2))**2,x)

[Out]

Integral((a + b*atanh(c/x**2))**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x^2))^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(c/x^2) + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (\frac {c}{x^2}\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x^2))^2,x)

[Out]

int((a + b*atanh(c/x^2))^2, x)

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